Let $R=\frac{\mathbb{Q}[t]}{(t^2-1)}$; trivially, $R$ is not an integral domain, since $(\overline{t-1})(\overline{t+1})=\overline{t^2-1}=\overline{0}$.

Is it possible to find $A,B,w \in R[x,y]$ such that the following three conditions are satisfied:

(i)$\operatorname{Jac}(A,B)\in \{1,-1\}$.(ii)$\operatorname{Jac}(A,w)=0$.(iii)$w \notin R[A]$.

**Remarks:**

Here we are not able to talk about normal or non-normal domains (in the simplest meaning), because now $R$ is not an integral domain.

Notice that $A=\overline{t−1}x+\overline{t}y$, $B=\overline{t}x+\overline{t+1}y$, $w=\overline{t+1}y$ do not satisfy all the three conditions; indeed, conditions

**(i)**and**(ii)**are satisfied, but condition**(iii)**is not, since $R[A] \ni \overline{t+1}A=\overline{t+1}(\overline{t−1}x+\overline{t}y)= \overline{t^2-1}x+\overline{(t+1)t}y=\overline{0}x+\overline{t^2+t}y=\overline{1+t}y=w$.

Thank you very much!